Phase noise
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In
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
, phase noise is the
frequency-domain In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a si ...
representation of random fluctuations in the
phase Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform * Phase space, a mathematic ...
of a
waveform In electronics, acoustics, and related fields, the waveform of a signal is the shape of its graph as a function of time, independent of its time and magnitude scales and of any displacement in time.David Crecraft, David Gorham, ''Electro ...
, corresponding to
time-domain Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the ca ...
deviations from perfect periodicity (
jitter In electronics and telecommunications, jitter is the deviation from true periodicity of a presumably periodic signal, often in relation to a reference clock signal. In clock recovery applications it is called timing jitter. Jitter is a significa ...
). Generally speaking,
radio-frequency Radio frequency (RF) is the oscillation rate of an alternating electric current or voltage or of a magnetic, electric or electromagnetic field or mechanical system in the frequency range from around to around . This is roughly between the upp ...
engineers speak of the phase noise of an
oscillator Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
, whereas digital-system engineers work with the jitter of a clock.


Definitions

Historically there have been two conflicting yet widely used definitions for phase noise. Some authors define phase noise to be the
spectral density The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
of a signal's phase only, while the other definition refers to the phase spectrum (which pairs up with the amplitude spectrum) resulting from the
spectral estimation In statistical signal processing, the goal of spectral density estimation (SDE) or simply spectral estimation is to estimate the spectral density (also known as the power spectral density) of a signal from a sequence of time samples of the signa ...
of the signal itself. Both definitions yield the same result at offset frequencies well removed from the carrier. At close-in offsets however, the two definitions differ. The
IEEE The Institute of Electrical and Electronics Engineers (IEEE) is a 501(c)(3) professional association for electronic engineering and electrical engineering (and associated disciplines) with its corporate office in New York City and its operation ...
defines phase noise as where the "phase instability" is the one-sided spectral density of a signal's phase deviation. Although is a one-sided function, it represents "the double-sideband spectral density of phase fluctuation". The symbol is called a ''(capital or uppercase) script L''.


Background

An ideal
oscillator Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
would generate a pure
sine wave A sine wave, sinusoidal wave, or just sinusoid is a curve, mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph of a function, graph. It is a type of continuous wave and also a Smoothness, smooth p ...
. In the frequency domain, this would be represented as a single pair of Dirac delta functions (positive and negative conjugates) at the oscillator's frequency; i.e., all the signal's
power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...
is at a single frequency. All real oscillators have
phase modulated Phase modulation (PM) is a modulation pattern for conditioning communication signals for transmission. It encodes a message signal as variations in the instantaneous phase of a carrier wave. Phase modulation is one of the two principal forms ...
noise Noise is unwanted sound considered unpleasant, loud or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference arise ...
components. The phase noise components spread the power of a signal to adjacent frequencies, resulting in noise
sidebands In radio communications, a sideband is a band of frequencies higher than or lower than the carrier frequency, that are the result of the modulation process. The sidebands carry the information transmitted by the radio signal. The sidebands com ...
. Oscillator phase noise often includes low frequency
flicker noise Flicker noise is a type of electronic noise with a 1/''f'' power spectral density. It is therefore often referred to as 1/''f'' noise or pink noise, though these terms have wider definitions. It occurs in almost all electronic devices and can show ...
and may include
white noise In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, ...
. Consider the following noise-free signal: : . Phase noise is added to this signal by adding a
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
represented by φ to the signal as follows: : . Phase noise is a type of
cyclostationary noise A cyclostationary process is a signal having statistical properties that vary cyclically with time. A cyclostationary process can be viewed as multiple interleaved stationary processes. For example, the maximum daily temperature in New York City ca ...
and is closely related to
jitter In electronics and telecommunications, jitter is the deviation from true periodicity of a presumably periodic signal, often in relation to a reference clock signal. In clock recovery applications it is called timing jitter. Jitter is a significa ...
, a particularly important type of phase noise is that produced by oscillators. Phase noise () is typically expressed in units of
dBc dBc (decibels relative to the carrier) is the power ratio of a signal to a carrier signal, expressed in decibels. For example, phase noise is expressed in dBc/ Hz at a given frequency offset from the carrier. dBc can also be used as a measurement o ...
/Hz, and it represents the noise power relative to the carrier contained in a 1 Hz bandwidth centered at a certain offsets from the carrier. For example, a certain signal may have a phase noise of −80 dBc/Hz at an offset of 10 kHz and −95 dBc/Hz at an offset of 100 kHz. Phase noise can be measured and expressed as single-sideband or double-sideband values, but as noted earlier, the IEEE has adopted the definition as one-half of the double-sideband PSD.


Jitter conversions

Phase noise is sometimes also measured and expressed as a power obtained by integrating over a certain range of offset frequencies. For example, the phase noise may be −40 dBc integrated over the range of 1 kHz to 100 kHz. This integrated phase noise (expressed in degrees) can be converted to jitter (expressed in seconds) using the following formula: :\text) = \frac In the absence of 1/f noise in a region where the phase noise displays a –20dBc/decade slope (
Leeson's equation Leeson's equation is an empirical expression that describes an oscillator's phase noise spectrum. Leeson's expression for single-sideband (SSB) phase noise in dBc/Hz (decibels relative to output level per hertz) and augmented for flicker noise: :L( ...
), the RMS cycle jitter can be related to the phase noise by: : \sigma^2_c = \frac Likewise: : \mathcal\left(f\right) = \frac


Measurement

Phase noise can be measured using a
spectrum analyzer A spectrum analyzer measures the magnitude of an input signal versus frequency within the full frequency range of the instrument. The primary use is to measure the power of the spectrum of known and unknown signals. The input signal that most co ...
if the phase noise of the device under test (DUT) is large with respect to the spectrum analyzer's
local oscillator In electronics, a local oscillator (LO) is an electronic oscillator used with a mixer to change the frequency of a signal. This frequency conversion process, also called heterodyning, produces the sum and difference frequencies from the frequenc ...
. Care should be taken that observed values are due to the measured signal and not the shape factor of the spectrum analyzer's filters. Spectrum analyzer based measurement can show the phase-noise power over many decades of frequency; e.g., 1 Hz to 10 MHz. The slope with offset frequency in various offset frequency regions can provide clues as to the source of the noise; e.g., low frequency
flicker noise Flicker noise is a type of electronic noise with a 1/''f'' power spectral density. It is therefore often referred to as 1/''f'' noise or pink noise, though these terms have wider definitions. It occurs in almost all electronic devices and can show ...
decreasing at 30 dB per decade (= 9 dB per octave). Phase noise measurement systems are alternatives to spectrum analyzers. These systems may use internal and external references and allow measurement of both residual (additive) and absolute noise. Additionally, these systems can make low-noise, close-to-the-carrier, measurements.


Spectral purity

The sinewave output of an ideal
oscillator Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
is a single line in the frequency spectrum. Such perfect spectral purity is not achievable in a practical oscillator. Spreading of the spectrum line caused by phase noise must be minimised in the local oscillator for a
superheterodyne receiver A superheterodyne receiver, often shortened to superhet, is a type of radio receiver that uses frequency mixing to convert a received signal to a fixed intermediate frequency (IF) which can be more conveniently processed than the original carr ...
because it defeats the aim of restricting the receiver frequency range by filters in the IF (intermediate frequency) amplifier.


See also

*
Allan variance The Allan variance (AVAR), also known as two-sample variance, is a measure of frequency stability in clocks, oscillators and amplifiers. It is named after David W. Allan and expressed mathematically as \sigma_y^2(\tau). The Allan deviation (ADE ...
*
Flicker noise Flicker noise is a type of electronic noise with a 1/''f'' power spectral density. It is therefore often referred to as 1/''f'' noise or pink noise, though these terms have wider definitions. It occurs in almost all electronic devices and can show ...
*
Leeson's equation Leeson's equation is an empirical expression that describes an oscillator's phase noise spectrum. Leeson's expression for single-sideband (SSB) phase noise in dBc/Hz (decibels relative to output level per hertz) and augmented for flicker noise: :L( ...
*
Maximum time interval error Maximum time interval error (MTIE) is the maximum error committed by a clock under test in measuring a time interval for a given period of time. It is used to specify clock stability requirements in telecommunications standards. MTIE measurements ...
*
Noise spectral density In communications, noise spectral density (NSD), noise power density, noise power spectral density, or simply noise density (''N''0) is the power spectral density of noise or the noise power per unit of bandwidth. It has dimension of power over ...
*
Spectral density The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
*
Spectral phase The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, o ...
*
Opto-electronic oscillator An opto-electronic oscillator (OEO) is an optoelectronic circuit that produces repetitive electronic sine wave and/or modulated optical continuous wave signals. An opto-electronic oscillator is based on converting the continuous light energy fr ...


References


Further reading

* * * * * * * * Ulrich L. Rohde, A New and Efficient Method of Designing Low Noise Microwave Oscillators, https://depositonce.tu-berlin.de/bitstream/11303/1306/1/Dokument_16.pdf * Ajay Poddar, Ulrich Rohde, Anisha Apte, “ How Low Can They Go, Oscillator Phase noise model, Theoretical, Experimental Validation, and Phase Noise Measurements”, IEEE Microwave Magazine, Vol. 14, No. 6, pp. 50–72, September/October 2013. * Ulrich Rohde, Ajay Poddar, Anisha Apte, “Getting Its Measure”, IEEE Microwave Magazine, Vol. 14, No. 6, pp. 73–86, September/October 2013 * U. L. Rohde, A. K. Poddar, Anisha Apte, “Phase noise measurement and its limitations”, Microwave Journal, pp. 22–46, May 2013 * A. K. Poddar, U.L. Rohde, “Technique to Minimize Phase Noise of Crystal Oscillators”, Microwave Journal, pp. 132–150, May 2013. * A. K. Poddar, U. L. Rohde, and E. Rubiola, “Phase noise measurement: Challenges and uncertainty”, 2014 IEEE IMaRC, Bangalore, Dec 2014. {{DEFAULTSORT:Phase Noise Oscillators Frequency-domain analysis Telecommunication theory Noise (electronics)